Does the position of the $1$ in a Jordan block matters?

Question

I am studying the Jordan Canonical Form of a matrix and I noticed that most of the books put the $1's$ of the Jordan blocks on the superdiagonal like this for example :

\begin{pmatrix} \lambda & 1 \\ 0 & \lambda \end{pmatrix}

But my professor puts it on the subdiagonal :

\begin{pmatrix} \lambda & 0 \\ 1& \lambda \end{pmatrix}

Does it make any difference ? I know this might change the position of the vectors of the Jordan basis on the matrix $P$ such that $J = P^{-1} A P$ , but can can I say that one of this constructions is better than the other or that one of them is more correct than the other ?

Answer 1

Applying the transpose operation doesn't change the trace, the determinant and the eigenvalues.

The only thing to consider is to do the operations that books do on rows onto columns and operations that books do on columns onto the rows of the matrix.

Both of the writing are correct equivalent, there is no "best", just that they are the transpose of each other $(AB)^T=(B^T A^T)$, just be careful with the orders of operation.

Answer 2

The two conventions are connected by writing the Jordan basis down in reverse order. Neither is better than the other.